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1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To)01-08-2006 FINAL Jan. 1, 2003- Dec. 31, 20054. TITLE AND SUBTITLE 5a. CONTRACT NUMBEROptimization-based robust nonlinear control

5b. GRANT NUMBERF49620-03-1-02035c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBERAndrew R. Teel

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORTNUMBER

Andrew R. TeelDepartment of Electrical andComputer EngineeringUniversity of CaliforniaSanta Barbara, CA 93106-95609. SPONSORING I MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)

Lt Col Sharon Heise,USAF,PhD Director, Math & Inf. Sci.AFOSR Program Manager, Dyn. & Control875 N. Randolph Street 11. SPONSOR/MONITOR'S REPORTSte 325, Rm 3112 NUMBER(S)Arlington, VA 2220312. DISTRIBUTION I AVAILABILITY STATEMENT

Unlimited /-\-eprvr-ed r

13. SUPPLEMENTARY NOTES

14. ABSTRACTNew control algorithms were developed for robust stabilization of nonlinear dynamicalsystems. Novel, linear matrix inequality-based synthesis algorithms were developed for theanti-windup problem. Results were extended so that they are applicable to all stabilizablelinear systems with input saturation. New insights into the closed-loop behavior of modelpredictive control were discovered. It was shown that certain model predictive controlalgorithms induce stability without any robustness. Then it was shown how these algorithmscan be modified to guarantee robustness. Formulas for horizon length to guarantee robuststabilization were given, and in the process it was shown that many of the standardassumptions in model predictive control could be relaxed. Extremum seeking controlalgorithms were advanced, and developed for systems with constraints and with nonsmoothresponse maps. Finally, some initial progress was made on the use of logical elements innonlinear control algorithms and the understanding of hybrid control systems in general.

15. SUBJECT TERMSNonlinear control and systems analysis

16. SECURITY CLASSIFICATION OF: 17. LIMITATION 18. NUMBER 19a. NAME OF RESPONSIBLE PERSONOF ABSTRACT OF PAGES Andrew R. Teel

a. REPORT b. ABSTRACT c. THIS PAGE 19b. TELEPHONE NUMBER (include areaUnclassified Unclassified Unclassified code)

(805) 893-3616

Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std. Z39.18

FINAL REPORT

OPTIMIZATION-BASED ROBUST NONLINEAR CONTROL

AFOSR F49620-03-1-0203

Submitted to the Air Force Office of Scientific ResearchDirectorate of Mathematical and Computer Sciences

August 1, 2006

PI: Andrew R. TeelCenter for Control Engineering and Computation

University of CaliforniaSanta Barbara, CA 93106-9560

20060809635

FINAL TECHNICAL REPORT:AFOSR Grant No. F49620-03-1-0203

Submitted toAir Force Office of Scientific Research

Arlington, VA 22203

Grant Title: Optimization-based robust nonlinear control

Starting Date: January 1, 2003

Ending Date: December 31, 2005

Report Date: August 1, 2006

Institution Name: University of CaliforniaSanta Barbara, CA 93106-9560

Principal Investigator: Andrew R. Teel, ProfessorDepartment of Electrical and Computer Engineering(805) 893-3616teel@ece.ucsb.edu

Business Office: UCSB Office of ResearchSanta Barbara, CA 93106

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Table of Contents

Report Documentation Pages 0Table of Contents 3Summary 4Research publications 5Research accomplishments 12Technology transitions or transfer 17Personnel 17Awards 17

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Summary

The objective of this research was to further advance optimization-based robust nonlinearcontrol design, for general nonlinear systems (especially in discrete time), for linearsystems with input and state constraints, and for systems that need to be positioned at theextremum of an unknown response surface. We accomplished our goals throughdevelopments: 1) in the area of anti-windup synthesis, where we provided new optimalsynthesis results using linear matrix inequalities, and provided new optimal synthesisresults using receding horizon optimal control solutions, 2) in the area of modelpredictive control for nonlinear systems, establishing stability under greatly weakenedassumptions on the optimization problem data, establishing that some existing modelpredictive control algorithms have no robustness margins, and establishing novel outputfeedback solutions, and 3) in the area of extremum seeking providing new algorithms toaccount for constraints and optimization on nonsmooth surfaces. The results of our workare described in thirty journal publications, three book chapters, and forty-eightconference papers.

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Research Publications

The research supported by this grant resulted in 30 published journal papers, 48 refereedconference papers, and 3 book chapters. The published papers are listed below.

Journal papers

1. "Conjugate convex Lyapunov functions for dual linear differential inclusions", R.Goebel, A.R. Teel, T. Hu and Z. Lin, IEEE Transactions on Automatic Control,vol. 51, no. 4, pp. 661-666, April 2006.

2. "Exremum seeking methods for optimization of variable cam timing engineroperation", D. Popovic, M. Jankovic, S. Magner, and A.R. Teel, IEEETransactions on Control Systems Technology, vol. 14, no. 3, p. 398-407, May2006.

3. "A unified framework for input-to-state stability in systems with two time scales",A.R. Teel, L. Moreau and D. Nesic, IEEE Transactions on Automatic Control,vol. 48, no. 9, pp. 1526-1544, September 2003.

4. "Anti-windup for stable linear systems with input saturation: an LMI-basedsynthesis", G. Grimm, J. Hatfield, I. Postlethwaite, A.R. Teel, M. Turner, L.Zaccarian, IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1509-1525, September 2003.

5. "Nonlinear Scheduled anti-windup design for linear systems", L. Zaccarian andA.R. Teel, IEEE Transactions on Automatic Control, vol. 49, no. 11, pp. 2055-2061, Novemeber 2004.

6. "Input-output stability properties of networked control systems", D. Nesic andA.R.Teel, IEEE Transactions on Automatic Control, vol. 49, no. 10, pp. 1650-1667, October 2004.

7. "Examples of GES systems that can be driven to infinity by arbitrarily smalladditive decaying exponentials", A.R. Teel and J. Hespanha, IEEE Transactionson Automatic Control, vol. 49, no. 8, pp. 1407-1410, August 2004.

8. "A framework for stabilization of nonlinear sampled-data systems based on theirapproximate discrete-time models", D. Nesic and A.R. Teel, IEEE Transactionson Automatic Control, vol. 49, no. 7, pp. 1103-1122, July 2004.

9. "Further results on robustness of (possibly discontinuous) sample and holdfeedback", C.M. Kellett, H. Shim and A. R. Teel, IEEE Transactions onAutomatic Control, vol. 49, no. 7, pp. 1081-1089, July 2004.

10. "Nonlinear antiwindup applied to Euler-Lagrange Systems", F. Morabito, A.R.Teel and L. Zaccarian, IEEE Trasnactions on Robotics and Automation, vol. 20,no. 3, pp. 526-537, June 2004.

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11. "Model predictive control: for want of a local control Lyapunov function, all isnot lost", G. Grimm, M. Messina, S.E. Tuna, and A.R. Teel, IEEE Transactionson Automatic Control, vol. 50, no. 5, pp. 546-558, May 2005.

12. "A nested Matrosov theorem and persistency of excitation for uniformconvergence in stable nonautonomous systems", A. Loria, E. Panteley, D.Popovic, and A.R. Teel, IEEE Transactions on Automatic Control, vol. 50, no. 2,pp. 183-198, February 2005.

13. "An anti-windup strategy for active vibration isolation systems", A.R. Teel, L.Zaccarian, and J. Marcinkowski, Control Engineering Practice, vol. 14, no. 1, pp.17-27, January 2006.

14. "L2 anti-windup for linear dead-time systems", L. Zaccarian, D. Nesic and A.R.Teel, Systems & Control Letters, vol. 54, no. 12, pp. 1205-1217, Decemer 2005.

15. "Conjugate Lyapunov functions for saturated linear systems", T. Hu, R. Goebel,A.R. Teel, and Z. Lin, Automatica, vol. 41, no. 11, pp. 1949-1956, November2005.

16. "Lyapunov characterizations of forced oscillations", T. Hu, A.R. Teel, and Z. Lin,Automatica, vol. 41, no. 10, pp. 1723-1735, October 2005.

17. "The L2 (12) bumpless transfer problem for linear plants: its definition andsolution", L. Zaccarian and A.R. Teel, Automatica, vol. 41, no. 7, pp. 1273-1280,July 2005.

18. "Discrete-time certainty equivalence output feedback: allowing discontinuouscontrol laws including those from model predictive control", M. Messina, S.E.Tuna, and A.R. Teel, Automatica, vol. 41, no. 4, pp. 617--628, April 2005.

19. "Input-to-state stability of networked control systems", D. Nesic and A.R. Teel,Automatica, vol. 40, no. 12, pp. 2121-2128, December 2004.

20. "Linear LMI-based external anti-windup augmentation for stable linear systems",G. Grimm, A.R. Teel and L. Zaccarian, Automatica, vol. 40, no. 11, pp. 1987--1996, November 2004.

21. "Examples when nonlinear model predictive control is nonrobust", G. Grimm, M.Messina, S.E. Tuna, and A.R. Teel, Automatica, vol. 40, no. 10, pp. 1729--1738,October 2004.

22. "Discrete-time asymptotic controllability implies smooth control-Lyapunovfunction", C.M. Kellett and A.R. Teel, Systems & Control Letters, vol. 52, no. 2,pp. 349-359, August 2004.

23. "Smooth Lyapunov functions and robustness of stability for differenceinclusions", C.M. Kellett and A.R. Teel, Systems & Control Letters, vol. 52, no. 2,pp. 395-405, August 2004.

24. "Matrosov theorem for parameterized families of discrete-time systems", D.Nesic and A.R. Teel, Automatica, vol. 40, no. 6, pp. 1025-1034, June 2004.

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25. "Anti-windup synthesis via sampled-data piecewise affine optimal control", A.Bemporad, A.R. Teel, and L. Zaccarian, Automatica, vol. 40, no. 4, pp. 549-562,April 2004.

26. "Input-to-state set stability for pulse width modulated control systems withdisturbances", A.R. Teel, L. Moreau and D. Nesic, Systems & Control Letters,vol. 51, no. 1, pp. 23-32, January 2004.

27. "On the robustness of KL-stability for difference inclusions: smooth discrete-timeLyapunov functions", C.M. Kellett and A.R. Teel, SIAM Journal on Control andOptimization, v. 44, no. 3, pp. 777-800, March 2005.

28. "Uniform parametric convergence in the adaptive control of mechanical systems",A. Loria, R. Kelly, and A.R. Teel, European J. of Control, vol. 11, no. 2, pp. 87-100, February 2005.

29. "Nonlinear anti-windup for manual flight control", C. Barbu, S. Galeani, A.R.Teel, and L. Zaccarian, International Journal of Control, vol. 78, no. 14, pp. 1111-1129, September 2005.

30. "Robust linear anti-windup synthesis for recovery of unconstrained performance",G. Grimm, A.R. Teel, L. Zaccarian, International Journal of Robust and NonlinearControl, vol. 14, no. 13-14, pp. 1133-1168, September 2004.

Conference papers

1. "Shorter horizons for model predictive control", S.E. Tuna, M. Messina and A.R.Teel, Proceedings of the 2006 American Control Conference, pp. 863--868,Minneapolis, MN, June 2006.

2. "Control of saturated linear plants via output feedback containing an internaldeadzone loop", Proceedings of the 2006 American Control Conference, pp. 5239--5244, Minneapolis, MN, June 2006.

3. "Experimental study of nonlinear anti-windup applied to an integrating linear plant",D. Orlando, L. Zaccarian, and A.R. Teel, Proceedings of the 2006 American ControlConference, pp. 5225--5226, Minneapolis, MN, June 2006.

4. "On input-to-state stability of impulsive systems", J.P. Hespanha, D. Liberzon, andA.R. Teel, Proc. 4 4th IEEE Conference on Decision and Control, pp. 4891-4896,Seville, Spain, December 2005.

5. "Regulating discrete-time homogeneous systems under arbitrary switching", S.E.Tuna and A.R. Teel, Proc. 4 4th IEEE Conference on Decision and Control, pp. 2586--2591, Seville, Spain, December 2005.

6. "On hybrid controllers that induce input-to-state stability with respect to measurementnoise", R.G. Sanfelice and A.R. Teel, Proc. 44h IEEE Conference on Decision andControl, pp. 4891--4896, Seville, Spain, December 2005.

7. "Results on input-to-state stability for hybrid systems", C. Cai and A.R. Teel, Proc.4 4th IEEE Conference on Decision and Control, pp. 5403--5408, Seville, Spain,

December 2005.

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8. "Results on robust stabilization of asymptotically controllable systems by hybridfeedback", C. Prieur, R. Goebel, and A.R. Teel, Proc. 44'ý IEEE Conference onDecision and Control, pp. 2598--2603, Seville, Spain, December 2005.

9. "Non-quadratic Lyapunov functions for performance analysis of saturated systems",T. Hu, A.R. Teel, and L. Zaccarian, Proc. 44'ý IEEE Conference on Decision andControl, pp. 8106--8111, Seville, Spain, December 2005.

10. "Performance analysis of saturated systems via two forms of differential inclusions",T. Hu, A.R. Teel, and L. Zaccarian, Proc. 444 IEEE Conference on Decision andControl, pp. 8100--8105, Seville, Spain, December 2005.

11. "Input-output stability of wireless networked control systems", M. Tabbara, D. Nesic,and A.R. Teel, Proc. 4 4th IEEE Conference on Decision and Control, pp. 209--214,Seville, Spain, December 2005.

12. "Regional anti-windup compensation for linear systems with input saturation", T. Hu,A.R. Teel, and L. Zaccarian, Proceedings of the 2005 American Control Conference,pp. 3397--3402, Portland, OR, June 2005.

13. "Nonlinear L2 gain and regional analysis for systems with anti-windupcompensation", T. Hu, A.R. Teel, and L. Zaccarian, Proceedings of the 2005American Control Conference, pp., Portland, OR, June 2005.

14. "First order reset elements and the Clegg integrator revisited", L. Zaccarian, D. Nesicand A.R. Teel, Proceedings of the 2005 American Control Conference, pp. 563--568,Portland, OR, June 2005.

15. "Results on solution sets to hybrid systems with application to stability theory", R.Goebel and A.R. Teel, Proceedings of the 2005 American Control Conference, pp.557--562, Portland, OR, June 2005.

16. "Converse Lyapunov theorems and robust asymptotic stability for hybrid systems",C. Cai, A.R. Teel and R. Goebel, Proceedings of the 2005 American ControlConference, pp. 12-17, Portland, OR, June 2005.

17. "Results on convergence in hybrid systems via detectability and an invarianceprinciple", R.G. Sanfelice, R. Goebel, and A.R. Teel, Proceedings of the 2005American Control Conference, pp. 551-556, Portland, OR, June 2005.

18. "L2 anti-windup for linear dead-time systems", D. Nesic, A.R. Teel and L. Zaccarian,Proceedings of the 2004 American Control Conference, pp. 5280-5285, Boston, MA,June/July 2004.

19. "The L2 (12) bumpless transfer problem: its definition and solution", L. Zaccarian andA.R. Teel, Proceedings of the 4 3 rd IEEE Conference on Decision and Control, pp.5505-5510, Bahamas, December 2004.

20. "Stability regions for saturated linear systems via conjugate Lyapunov functions", T.rd

Hu, Z. Lin, R. Goebel, and A.R. Teel, Proceedings of the 43 IEEE Conference onDecision and Control, pp. 5499-5504, Bahamas, December 2004.

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21. "Generalized dilations and numerically solving discrete-time homogeneousoptimization problems", S.E. Tuna and A.R. Teel, Proceedings of the 4 3 rd IEEEConference on Decision and Control, pp. 5403--5408, Bahamas, December 2004.

22. "Anti-windup for marginally stable plants applied to open water channels", L.Zaccarian, Y. Li, E. Weyer, M. Cantoni and A.R. Teel, Proceedings of the th AsianControl Conference, pp. 1692--1700, Melbourne, Australia, July 2004.

23. "Dissipativity for dual linear differential inclusions through conjugate storagefunctions", R. Goebel, A.R. Teel, T. Hu, and Z. Lin, Proceedings of the 43d IEEEConference on Decision and Control, pp. 2700--2705, Bahamas, December 2004.

24. "Discrete-time homogeneous Lyapunov functions for homogeneous differenceinclusions", S.E. Tuna and A.R. Teel, Proceedings of the 4 3rd IEEE Conference onDecision and Control, pp. 1606--1610, Bahamas, December 2004.

25. "Direct search methods for nonsmooth optimization", D. Popovic and A.R. Teel,Proceedings of the 4 3rd IEEE Conference on Decision and Control, pp. 3173--3178,Bahamas, December 2004.

26. "Constructive nonlinear anti-windup design for exponentially unstable linearsystems", S. Galeani, A.R. Teel, and L. Zaccarian, Proceedings of the 4 3rd IEEEConference on Decision and Control, pp. 5028--5033, Bahamas, December 2004.

27. "On performance and robustness issues in the anti-windup problem", S. Galeani andA.R. Teel, Proceedings of the 4 3rd IEEE Conference on Decision and Control, pp.5022-5033, Bahamas, December 2004.

28. "A note of input-to-state stability of networked control systems", D. Nesic and A.R.Teel, Proceedings of the 43d IEEE Conference on Decision and Control, pp. 4613--4618, Bahamas, December 2004.

29. "A Lyapunov approach to frequency analysis", T. Hu, A.R. Teel, and Z. Lin,Proceedings of the 2004 American Control Conference, pp. 4145-4150, Boston,MA, June/July 2004.

30. "Robust stabilization of discrete-time nonlinear systems by certainty equivalenceoutput feedback with applications to model predictive control", M. Messina, S.E.Tuna, A.R. Teel, Proceedings of the 2004 American Control Conference, pp. 2202--2207, Boston, MA, June/July 2004.

31. "Maneuvering dynamical systems by sliding-mode control", R. Skjetne and A.R.Teel, Proceedings of the 2004 American Control Conference, pp. 1277--1282,Boston, MA, June/July 2004.

32. "Results on discrete-time control-Lyapunov functions", C.M. Kellett and A.R. Teel,Proceedings of the 2004 American Control Conference, pp. 5961--5966, Boston, MA,June/July 2004.

33. "Nominally robust model predictive control with state constraints", G. Grimm, M.Messina, S.E. Tuna and A.R. Teel, Proceedings of the 2004 American ControlConference, pp. 1413--1418, Boston, MA, June/July 2004.

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34. "A Matrosov theorem with an application to model reference adaptive control viaapproximate discrete-time models", D. Nesic and A.R. Teel, Proceedings of the 4 2nd

IEEE Conference on Decision and Control, pp. 3627-3632, Maui, HI, December2003.

35. "Results on converse Lyapunov theorems for difference inclusions", C.M. Kellett andA.R. Teel, Proceedings of the 42 d IEEE Conference on Decision and Control, pp.3627--3632, Maui, HI, December 2003.

36. "Examples of zero robustness in constrained model predictive control", G. Grimm,M. Messina, S.E. Tuna and A.R. Teel, Proceedings of the 4 2nd IEEE Conference onDecision and Control, pp. 3724--3729, Maui, HI, December 2003.

37. "Establishing Lipschitz properties of multivariable algebraic loops with incrementalsector nonlinearities", G. Grimm, A.R. Teel and L. Zaccarian, Proceedings of the 4 2nd

IEEE Conference on Decision and Control, pp. 5667--5672, Maui, HI, December2003.

38. "Nonlinear stabilization of a spherical particle trapped in an optical tweezer", A.Ranaweera, B. Bamieh, and A.R. Teel, Proceedings of the 4 2 nd IEEE Conference onDecision and Control, pp. 3431--3436, Maui, HI, December 2003.

39. "Lp stability of networked control systems", D. Nesic and A.R. Teel, Proceedings ofnd

the 42d IEEE Conference on Decision and Control, pp. 1188--1193, Maui, HI,December 2003.

40. "Uniform parametric convergence in the adaptive control of manipulators: a caserestudied", A. Loria, R. Kelly, and A.R. Teel, Proceedings of the 2003 IEEEInternational Conference on Robotics and Automation, pp. 1062-1067, Taipei,Taiwan, September 2003.

41. "Extremum seeking methods for optimization of variable cam timing engineoperation", D. Popovic, M. Jankovic, S. Magner and A. Teel, Proceedings of the2003 American Control Conference, pp. 3136--3141, Denver, CO, June 2003.

42. "The 12 anti-windup problem for discrete-time linear systems: definition andsolutions", G. Grimm, A.R. Teel and L. Zaccarian, Proceedings of the 2003 AmericanControl Conference, pp. 5329-5334, Denver, CO, June 2003.

43. "Model predictive control when a local control Lyapunov function is not available",G. Grimm, M. Messina, A.R. Teel and S.E. Tuna, Proceedings of the 2003 AmericanControl Conference, pp. 4125--4130, Denver, CO, June 2003.

44. "An improved SPSA algorithm for stochastic optimization with bound constraints",D. Popovic, A.R. Teel, and M. Jankovic, Proceedings of the 1 6th IFAC WorldCongress, Prague, CZ, July 2005 (CD-ROM)

45. "Output feedback compensators for weakened anti-windup of additively perturbedsystems", S. Galeani, A.R. Teel and L. Zaccarian, Proceedings of the 1 6 th IFACWorld Congress, Prague, CZ, July 2005 (CD-ROM)

46. "Stability properties of reset systems", L. Zaccarian, D. Nesic, and A.R. Teel,Proceedings of the 1 6th IFAC World Congress, Prague, CZ, July 2005 (CD-ROM)

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47. "Generalized dilations and homogeneity", S.E. Tuna and A.R. Teel, Proceedings ofthe 1 6 th IFAC World Congress, Prague, CZ, July 2005 (CD-ROM)

48. "Hybrid MPC: open-minded but not easily swayed", S.E. Tuna, R.G. Sanfelice, M.J.Messina, and A.R. Teel, Proceedings of the International Workshop on Assessmentand Future Directions of Nonlinear Model Predictive Control, Freudenstadt-Lauterbad, Germany, August 2005, pp. 169-180.

49. "Nonlinear anti-windup for exponentially unstable linear plants", S. Galeani, A.R.Teel, and L. Zaccarian, In Current Trends in Nonlinear Systems and Control, Menini,Zaccarian and Abdallah, eds., pp. 143-162, Birkhauser, Boston, MA, 2006.

50. "Dual matrix inequalities in stability and performance analysis of lineardifferential/difference inclusions", R. Goebel, T. Hu, and A.R. Teel, In CurrentTrends in Nonlinear Systems and Control, Menini, Zaccarian and Abdallah, eds.,pp.103-122, Birkhauser, Boston, MA, 2006.

51. "On the literature's two different definitions of uniform global asymptotic stabilityfor nonlinear systems", A.R. Teel and L. Zaccarian, In Advanced Topics in ControlSystems Theory, Lecture Notes in Control and Information Sciences, 328, Loria,Lamnabhi-Lagarrigue, and Panteley, eds., pp. 285-289, Springer, London, 2006.

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Research Accomplishments

Anti-windup synthesis

The anti-windup problem is a control synthesis problem in the presence of inputsaturation. Its distinguishing feature is that a controller has already been designed and itmust operate without modification unless the actuators saturate. In this case, because theperformance of this controller might be poor in the presence of saturation, anti-windupaugmentation should activate to maintain stability and reasonable performance, and torestore closed-loop operation to the original controller when this becomes possible.Systematic solutions to this industry-motivated problem (including the motivation toeliminate pilot/flight control system-induced oscillations) have started to appear over thelast decade, and in the course of this grant we contributed general, systematic solutions ofvarious kinds.

Receding horizon optimal control-based synthesis: We demonstrated the highperformance capabilities of discrete-time disturbance rejection model predictive controlin the anti-windup problem. We exploited its efficient off-line synthesis, and have usedit for anti-windup synthesis in closed-loops that use high performance continuous-timecontrollers that ignore input saturation.

Euler-Lagrange systems: While anti-windup is usually conceived for linear controlsystems with input saturation, we showed under a previous AFOSR grant that it can alsobe posed and solved for more general nonlinear feedback loops. For the current grant, weapplied our earlier ideas, with modifications aimed at improved performance, to Euler-Lagrange systems including robotic manipulators such as the SCARA and PUMA. Wedemonstrated the superior performance of these anti-windup algorithms compared tostandard industrial practice which involves path planning to avoid input saturation or ad-hoc anti-windup strategies.

Systems with time delays: We presented new results on anti-windup for control systemswith time-delays. These results have the potential for significant applications innetworked control systems and control of networks.

Regional anti-windup performance: Our initial LMI-based anti-windup synthesisalgorithms only applied to closed-loop systems having plants that are open-loopexponentially stable. However, by the end of the grant period, we had developed LMI-based synthesis algorithms that were applicable for any closed-loop system. In additionto synthesis algorithms, we provided new algorithms for the analysis of existing anti-windup compensation. Our analysis algorithms, which permit the use of nonquadraticLyapunov functions, are not convex in general but our experience suggests that thealgorithms remain numerically tractable for problems of reasonable size. With thisincreased applicability, anti-windup algorithms had matured to the state where they areready to be summarized in a textbook. We have nearly completed such an effort in thecourse of this grant.

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Robustness, and modeling as differential inclusions: We continued to investigate therobustness of anti-windup control algorithms and elucidated some natural tradeoffsbetween performance and robustness in the anti-windup problem that arise from the meredefinition of the anti-windup problem. Moreover, we provided some relaxed problemdefinitions that allow stronger robustness properties. Finally, since input saturation isoften modeled as a general sector constraint leading to a linear differential inclusion, wehave developed a duality theory for linear differential inclusions, relying on the convexconjugate of a Lyapunov function to provide a Lyapunov function for the lineardifferential inclusion's dual. This result has significance for numerical investigations ofstability for closed-loops with input saturation, since some numerical algorithms (mostthat are based on nonquadratic Lyapunov functions) give very different answers forstability ranges when applied to the primal problem compared to results when applied tothe dual. In addition to these results, we developed a Lyapunov-based frequencyresponse theory for linear differential inclusions.

Bumpless transfer: A problem closely related to anti-windup is that of bumpless transfer.The objective of this problem is to provide control logic that enables a smooth transitionfrom one controller (perhaps manual control) to another controller. In the course of thegrant, we gave the first formal definition of this problem and provided a systematicsolution.

Model predictive control

Model predictive control (MPC) is an increasingly popular approach used to synthesizestabilizing feedback control laws for nonlinear systems. In industry, this control methodtypically is based on discrete-time models, which makes determining the MPC feedbacklaw equivalent to solving a finite dimensional optimization problem.' The latter often canbe carried out on-line, especially for slow plants like those in the chemical processindustry. Even for faster plants, including aero-vehicles, MPC is beginning to beconsidered as a reasonable feedback control alternative.

One of the main challenges in nonlinear MPC is making sure that the finite dimensionaloptimization problem leads to a stabilizing feedback. Significant attention has beengiven to this problem over the last fifteen years. The typical solution involves imposingstrict terminal constraints and/or limiting the functions used in the optimization problem.One issue that has received very little attention is the nominal robustness of these MPCschemes. In the course of this grant, we showed that some of the standard algorithms arecapable of producing closed-loop asymptotic stability with absolutely no robustness. Thelack of robustness enters due to state constraints coupled with the use of short horizonswhen the state constraints are terminal constraints used to induce stability. In all caseswhere the closed-loop has no robustness, the MPC feedback law is discontinuous, as isthe Lyapunov function used to establish asymptotic stability. In our work on this grant,we have showed how to distinguish robustness from zero robustness in discontinuousdiscrete-time systems. Equipped with these tools, we set out to reformulate MPCalgorithms so that robustness is guaranteed. In the process, we aimed to provide as muchflexibility as possible in the functions that can be used in the optimization problem.

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In our first contribution, we addressed unconstrained MPC and established new resultsrelaxing many of the typical requirements on the functions used in the MPC optimizationproblem. These relaxations came at the price of requiring sufficiently long horizons, andhence sufficiently large dimension, in the MPC optimization problem. However, wedemonstrated constructively on interesting examples that these sufficiently long horizonsmight actually be reasonably short. These results were enabled by careful exploitation ofvarious aspects of nonlinear systems theory including homogeneity theory.

In our second significant contribution, we addressed constrained MPC extending ourearlier results to the problem with inherent state constraints and showing such MPCalgorithms can be configured to guarantee nominal robustness. In the process, this workshowed that the use of terminal constraints does not necessarily lead to the absence ofrobustness if sufficiently long horizons are used.

We also showed how nonlinear observability properties can be exploited to implementMPC using output feedback. The only results of this type in the literature of which weare aware have made very non-generic assumptions about the Lipschitz continuity of theMPC feedback law. We demonstrated that this condition is not necessary forimplementing MPC by output feedback. This result makes it clear that MPC can bereadily applied to nonlinear control systems with limited measurements.

We also showed how to use the MPC idea for homogeneous, switching control systemsto develop feedback laws that induce stability that is robust to arbitrary switching. Weprovided a constructive algorithm that is realizable numerically for systems of low order.

Finally, we developed versions of model predictive control that employ logic variables toassist with the task of making robust decisions. We demonstrated the effectiveness ofthese algorithms in obstacle avoidance problems involving noisy measurements forhighly maneuverable air vehicles.

Extremum Seeking

Extremum Seeking (ES) is an iterative optimization process performed in real time on aphysical system that has a significant dynamical aspect. The derivatives of the function tobe optimized - system performance in steady state - are not available, but the functioncan be measured, modulo measurement noise and transients. ES controllers, usually builtaround numerical optimization algorithms, perform optimization by monitoring thesystem performance, and adjusting the parameters on-line to improve that performance.In our research we studied applicable numerical algorithms, analyzed the dynamicalinteraction between an optimization algorithm and the optimized dynamical system, andderived some recipes to achieve faster extremum seeking. We worked with Ford MotorCompany to accelerate optimization and calibration of automotive engines. We havedemonstrated advanced multi-variable extremum seeking methods in performanceoptimization for a dual-independent variable cam timing engine and have developed aninteractive software product that a non-ES specialist can use for this task.

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In work during this grant, we considered extremum seeking problems where the optimaloperating condition resides on a constraint boundary. In this case, we showed that thenafve application of certain numerical optimization algorithms, such as simultaneousperturbation stochastic approximation (SPSA) with projection, may exhibit especiallyslow convergence because of the interaction between the projection operator and agradient approximation. We then showed how to modify the standard SPSA algorithmto remove this effect and restore fast convergence that can be used in extremum seeking.We have also established new convergence properties using direct search algorithms fornonsmooth optimization problems. These results give a firm theoretical foundation forusing direct search within an extremum seeking framework.

Analysis of Networked Control Systems

It is common in modem control applications to find control loops that are closed via aserial communication channel that transmits signals from many sensors and actuators inthe system, as well as signals from other unrelated users that are connected to a network.Motivation for using this set-up comes from lower cost, ease of maintenance, greatflexibility, as well as low weight and volume. The main issue in these "NetworkedControl Systems" is that the serial communication channel has many "nodes" (sensorsand actuators) where only one node can report its value at a time and, hence, access to thechannel needs to be scheduled in an appropriate manner for a proper operation of thecontrol system. During this grant, we contributed to the stability analysis of nonlinearnetworked control systems. Based on the small gain theorem, we provided estimates forthe maximum allowable transfer interval (MATI) in networked control systems that aretypically multiple orders of magnitude less conservative than previous results existing inthe literature. Along the way, we have developed a useful paradigm for the classificationof networked control system protocols that is based on Lyapunov stability theory. Weexpect this characterization of efficient network protocols to lead to the development ofnew protocols that allow for larger MATIs.

Hybrid systems: control and analysis

Hybrid systems provide an exciting area of research strongly related to nonlinear control.A strong, foundational understanding of hybrid dynamical systems will allow bringingnonlinear control design tools to bear on a much wider class of systems, and will alsoenable systematically introducing logic-based switching into nonlinear control algorithmdesign. Researchers have struggled over the last decade to come up with a concisecharacterization of hybrid systems and their properties. Inspired by earlier work in thisarea, we began to develop an array of results that we believe will help to make hybridsystems more tractable. For example, we provided new results on convergence propertiesof solutions to hybrid systems, and related these properties to robust asymptotic stability.We established a general invariance principle in the spirit of LaSalle's invarianceprinciple for differential and difference equations. Also, we proved that asymptoticallystable hybrid systems admit smooth Lyapunov functions. We used these results as

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motivation to provide new Lyapunov proofs for the stability of networked controlsystems (both wired and wireless) and for control loops with reset elements, like the so-called "Clegg integrator". We investigated the degree to which the well-known "input-to-state" stability theory for differential equations carries over to hybrid systems. Thereare some subtle differences that appear in the hybrid domain. Nevertheless, we were ableto use hybrid control to induce input-to-state stability with respect to measurement noisewhere this is impossible to do with pure state feedback. We also gave novel conditionsfor input-to-state stability in impulsive dynamical systems. As an alternative to thesample and hold solutions that guarantee asymptotic stabilization that is robust tomeasurement noise for general, asymptotically controllable nonlinear systems, weshowed how this property can be induced for the same class of systems using logic-basedhysteresis switching. This same idea can be used to give a simple, systematic solution tocontrol problems like global swing-up of the inverted pendulum.

These results are poised to enable important new breakthroughs both in the use of hybridcontrol for nonlinear systems, and also the development of control algorithms for hybridsystems. These are some of the topics that we are considering in our subsequent work,which is also being supported by AFOSR.

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Personnel Supported

Faculty: Dr. Andrew R. Teel (PI);Graduate Student Researchers: Gene Grimm, Dobrivoje Popovic, Diane Dai

Transitions

Customer: Ford Research Laboratory, Dearborn, MIContact: Dr. Mrdjan JankovicAddress: P.O. Box 2053, MD 2036 SRL, 2101 Village Rd., Dearborn, MI 48121Telephone: (313) 390-8916.

Theory transitioned: Exremum seeking in dynamical systems

Application: Rapid engine calibration for a variable cam timing engine.

Plenary Lectures during period

IFAC Nonlinear Control Systems Design Symposium, Stuttgart, Germany, 2004.Asian Control Conference, Melbourne, Australia, 2004.

Lifetime honors/awards

IEEE Fellow, 2002AACC Donald P. Eckman Award, 1999SIAM Control and Systems Theory Prize, 1998

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